QLunch: Arthur Morris
Speaker: Arthur Morris
Title: Constraints on phantom codes from automorphism group bounds
Abstract: I will discuss the recently proposed phantom codes, which are defined by the property that every in-block logical CNOT circuit can be implemented with a physical permutation, a feature that has the potential to greatly reduce the depth of compiled circuits. I will show that phantomness comes at the cost of a low encoding rate. Specifically, any binary phantom code encoding k logical qubits into n physical qubits with distance d ≥ 2 obeys the bound k ≤ log2(n+1) for all k ≠ 4. This logarithmic ceiling cannot be circumvented by permitting additional local unitary gates or by making use of subsystem codes: any subspace or subsystem code admitting a SWAP-transversal implementation of every logical CNOT circuit is constrained to satisfy the same bound. These bounds follow from a general theorem relating the length of a quantum code to the structure of its automorphism group. I will show that, within the class of stabiliser codes with k ≠ 4, there is a unique family of codes saturating this bound. Finally, I will discuss the single exception to this constraint: for k = 4, there is a non-stabiliser ((8, 2^4, 2)) phantom code that violates the bound and has a transversal non-Clifford gate.